MATRIX TRACE INEQUALITIES ON THE TSALLIS ENTROPIES

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Matrix Trace Inequalities on the Tsallis Entropies

Shigeru Furuichi vol. 9, iss. 1, art. 1, 2008

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MATRIX TRACE INEQUALITIES ON THE TSALLIS ENTROPIES

SHIGERU FURUICHI

Department of Electronics and Computer Science Tokyo University of Science

Yamaguchi, Sanyo-Onoda City Yamaguchi, 756-0884, Japan EMail:furuichi@ed.yama.tus.ac.jp

Received: 28 June, 2007

Accepted: 29 January, 2008

Communicated by: F. Zhang

2000 AMS Sub. Class.: 47A63, 94A17, 15A39.

Key words: Matrix trace inequality, Tsallis entropy, Tsallis relative entropy and maximum entropy principle.

Abstract: Maximum entropy principles in nonextensive statistical physics are revisited as an application of the Tsallis relative entropy defined for non-negative matrices in the framework of matrix analysis. In addition, some matrix trace inequalities related to the Tsallis relative entropy are studied.

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Close Acknowledgements: The author would like to thank the reviewer for providing valuable com-

ments to improve the manuscript. The author would like to thank Professor K.Yanagi and Professor K. Kuriyama for providing valuable comments and constant encouragement. This work was supported by the Japanese Ministry of Education, Science, Sports and Culture, Grant-in-Aid for En- couragement of Young Scientists (B), 17740068. This work was also par- tially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (B), 18300003.

Dedicatory: Dedicated to Professor Kunio Oshima on his 60th birthday.

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1. Introduction

In 1988, Tsallis introduced the one-parameter extended entropy for the analysis of a physical model in statistical physics [10]. In our previous papers, we studied the properties of the Tsallis relative entropy [5, 4] and the Tsallis relative operator entropy [17,6]. The problems on the maximum entropy principle in Tsallis statistics have been studied for classical systems and quantum systems [9, 11, 2, 1]. Such problems were solved by the use of the Lagrange multipliers formalism. We give a new approach to such problems, that is, we solve them by applying the non- negativity of the Tsallis relative entropy without using the Lagrange multipliers formalism. In addition, we show further results on the Tsallis relative entropy.

In the present paper, the set of n× n complex matrices is denoted by M_n(C).

That is, we deal withn×n matrices because of Lemma2.2in Section2. However some results derived in the present paper also hold for the infinite dimensional case.

In the sequel, the set of all density matrices (quantum states) is represented by D_n(C)≡ {X ∈M_n(C) :X ≥0,Tr[X] = 1}.

X ∈ M_n(C)is called by a non-negative matrix and denoted byX ≥ 0, if we have hXx, xi ≥ 0for allx ∈ Cⁿ. That is, for a Hermitian matrixX, X ≥ 0means that all eigenvalues ofXare non-negative. In addition,X ≥Y is defined byX−Y ≥0.

For−I ≤ X ≤ I andλ ∈ (−1,0)∪(0,1), we denote the generalized exponential function byexp_λ(X)≡(I+λX)^1/λ. As the inverse function ofexp_λ(·), forX ≥0 andλ∈(−1,0)∪(0,1), we denote the generalized logarithmic function byln_λX ≡

X^λ−I

λ . Then the Tsallis relative entropy and the Tsallis entropy for non-negative matricesX andY are defined by

D_λ(X|Y)≡Tr

X^1−λ(ln_λX−ln_λY)

, S_λ(X)≡ −D_λ(X|I).

These entropies are generalizations of the von Neumann entropy [16] and of the

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Umegaki relative entropy [14] in the sense that

λ→0limS_λ(X) =S₀(X)≡ −Tr[XlogX]

and

λ→0limD_λ(X|Y) = D₀(X|Y)≡Tr[X(logX−logY)].

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2. Maximum Entropy Principle in Nonextensive Statistical Physics

In this section, we study the maximization problem of the Tsallis entropy with a constraint on theλ-expectation value. In quantum systems, the expectation value of an observable (a Hermitian matrix) H in a quantum state (a density matrix) X ∈ Dn(C)is written asTr[XH]. Here, we consider theλ-expectation valueTr[X^1−λH]

as a generalization of the usual expectation value. Firstly, we impose the following constraint on the maximization problem of the Tsallis entropy:

Cf_λ ≡

X ∈D_n(C) : Tr[X^1−λH] = 0 ,

for a givenn×nHermitian matrixH. We denote a usual matrix norm byk·k, namely forA∈Mn(C)andx∈Cⁿ,

kAk ≡ max

kxk=1kAxk. Then we have the following theorem.

Theorem 2.1. Let Y = Z_λ⁻¹exp_λ(−H/kHk), where Zλ ≡ Tr[exp_λ(−H/kHk)], for an n × n Hermitian matrix H and λ ∈ (−1,0) ∪(0,1). If X ∈ Cf_λ, then S_λ(X)≤ −c_λln_λZ_λ⁻¹,wherec_λ ≡Tr[X^1−λ].

Proof. SinceZ_λ ≥ 0 and we have ln_λ(x⁻¹Y) = ln_λY + (ln_λx⁻¹)Y^λ for a non- negative matrixY and scalarx, we calculate

Tr[X^1−λln_λY] = Tr[X^1−λln_λ

Z_λ⁻¹exp_λ(−H/kHk) ]

= Tr[X^1−λ

−H/kHk+ ln_λZ_λ⁻¹(I−λH/kHk) ]

= Tr[X^1−λ

lnλZ_λ⁻¹I−Z_λ^−λH/kHk ] =cλlnλZ_λ⁻¹,

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since ln_λZ_λ⁻¹ = ^Z

−λ λ −1

λ by the definition of the generalized logarithmic function ln_λ(·). By the non-negativity of the Tsallis relative entropy:

(2.1) Tr[X^1−λln_λY]≤Tr[X^1−λln_λX], we have

S_λ(X) = −Tr[X^1−λln_λX]≤ −Tr[X^1−λln_λY] =−c_λln_λZ_λ⁻¹.

Next, we consider the slightly changed constraint:

C_λ ≡

X ∈D_n(C) : Tr[X^1−λH]≤Tr[Y^1−λH] and Tr[X^1−λ]≤Tr[Y^1−λ] for a givenn×n Hermitian matrixH, as the maximization problem for the Tsallis entropy. To this end, we prepare the following lemma.

Lemma 2.2. For a givenn×nHermitian matrixH, ifnis a sufficiently large integer, then we haveZλ ≥1.

Proof.

(i) For a fixed0< λ <1and a sufficiently largen, we have

(2.2) (1/n)^λ ≤1−λ.

From the inequalities− kHkI ≤H ≤ kHkI, we have (2.3) (1−λ)^λ¹I ≤exp_λ(−H/kHk)≤(1 +λ)¹^λI.

By inequality (2.2), we have 1

nI ≤(1−λ)¹^λI ≤exp_λ(−H/kHk), which impliesZλ ≥1.

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(ii) For a fixed−1< λ <0and a sufficiently largen, we have

(2.4) (1/n)^λ ≥1−λ.

Analogously to (i), we have inequalities (2.3) for−1 < λ < 0. By inequality (2.4), we have

1

nI ≤(1−λ)¹^λI ≤exp_λ(−H/kHk), which impliesZ_λ ≥1.

Then we have the following theorem by the use of Lemma2.2.

Theorem 2.3. Let Y = Z_λ⁻¹exp_λ(−H/kHk), where Z_λ ≡ Tr[exp_λ(−H/kHk)], for λ ∈ (−1,0)∪(0,1) and ann ×n Hermitian matrix H. If X ∈ C_λ andn is sufficiently large, thenS_λ(X)≤S_λ(Y).

Proof. Due to Lemma2.2, we haveln_λZ_λ⁻¹ ≤ 0for a sufficiently largen. Thus we haveln_λZ_λ⁻¹Tr[X^1−λ] ≥ ln_λZ_λ⁻¹Tr[Y^1−λ]forX ∈ C_λ. Similarly to the proof of Theorem2.1, we have

Tr[X^1−λlnλY] = Tr[X^1−λlnλ

= Tr[X^1−λ

−H/kHk+ lnλZ_λ⁻¹(I−λH/kHk) ]

= Tr[X^1−λ

lnλZ_λ⁻¹I−Z_λ^−λH/kHk ]

≥Tr[Y^1−λ

ln_λZ_λ⁻¹I−Z_λ^−λH/kHk ]

= Tr[Y^1−λ

−H/kHk+ ln_λZ_λ⁻¹(I−λH/kHk) ]

= Tr[Y^1−λln_λ

= Tr[Y^1−λln_λY].

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By Eq.(2.1) we have

S_λ(X) =−Tr[X^1−λln_λX]≤ −Tr[X^1−λln_λY]≤ −Tr[Y^1−λln_λY] =S_λ(Y).

Remark 1. Since−x^1−λln_λxis a strictly concave function, S_λ is a strictly concave function on the setC_λ. This means that the maximizing Y is uniquely determined so that we may regardY as a generalized Gibbs state, since an original Gibbs state e^−βH/Tr[e^−βH], whereβ ≡1/T andT represents a physical temperature, gives the maximum value of the von Neumann entropy. Thus, we may define a generalized Helmholtz free energy by

F_λ(X, H)≡Tr[X^1−λH]− kHkS_λ(X).

This can be also represented by the Tsallis relative entropy such as F_λ(X, H) = kHkD_λ(X|Y) + ln_λZ_λ⁻¹Tr[X^1−λ(kHk −λH)].

The following corollary easily follows by taking the limit asλ→0.

Corollary 2.4 ([12,15]). LetY =Z₀⁻¹exp (−H/kHk), whereZ₀ ≡Tr[exp (−H/kHk)], for ann×nHermitian matrixH.

(i) IfX ∈fC₀, thenS₀(X)≤logZ₀. (ii) IfX ∈C₀, thenS₀(X)≤S₀(Y).

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3. On Some Trace Inequalities Related to the Tsallis Relative Entropy

In this section, we consider an extension of the following inequality [8]:

(3.1) Tr[X(logX+ logY)]≤ 1

pTr[XlogX^p/2Y^pX^p/2] for non-negative matricesX andY, andp >0.

For the proof of the following Theorem3.3, we use the following famous inequalities.

Lemma 3.1 ([8]). For any Hermitian matricesAandB,0 ≤λ ≤ 1andp > 0, we have the inequality:

Trh

e^pA]_λe^pB1/pi

≤Tr

e^{(1−λ)A+λB} ,

where theλ-geometric mean for positive matricesAandB is defined by A]_λB ≡A^1/2 A^−1/2BA^−1/2^λ

A^1/2.

Lemma 3.2 ([7,13]). For any Hermitian matrices GandH, we have the Golden- Thompson inequality:

Tr e^G+H

≤Tr e^Ge^H

.

Theorem 3.3. For positive matricesX andY,p≥1and0< λ≤1, we have (3.2) D_λ(X|Y)≤ −Tr[Xln_λ(X^−p/2Y^pX^−p/2)^1/p].

Proof. First of all, we note that we have the following inequality [3]

(3.3) Tr[(Y^1/2XY^1/2)^rp]≥Tr[(Y^r/2X^rY^r/2)^p]

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for non-negative matricesX and Y, and0 ≤ r ≤ 1, p > 0. Similar to the proof of Theorem 2.2 in [5], inequality (3.2) easily follows by setting A = logX and B = logY in Lemma3.1such that

Tr[(X^p]_λY^p)^1/p]≤Tr[e^log^X^1−λ^+log^Y^λ]

≤Tr[e^log^X^1−λe^log^Y^λ]

= Tr[X^1−λY^λ], (3.4)

by Lemma3.2. In addtion, we have

(3.5) Tr[X^rY^r]≤Tr[(Y^1/2XY^1/2)^r], (0≤r ≤1), on takingp= 1of inequality (3.3). By (3.4) and (3.5) we obtain:

Tr[(X^p]λY^p)^1/p] = Tr h

X^p/2(X^−p/2Y^pX^−p/2)^λX^p/2 ^1/p i

≥Tr[X(X^−p/2Y^pX^−p/2)^λ/p].

Thus we have,

D_λ(X|Y) = Tr[X−X^1−λY^λ] λ

≤ Tr[X−X(X^−p/2Y^pX^−p/2)^λ/p] λ

=−Tr[X

((X^−p/2Y^pX^−p/2)^1/p)^λ−I ] λ

=−Tr[Xln_λ(X^−p/2Y^pX^−p/2)^1/p].

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Remark 2. For positive matricesXandY,0< p <1and0< λ≤1, the following inequality does not hold in general:

(3.6) D_λ(X|Y)≤ −Tr[Xln_λ(X^−p/2Y^pX^−p/2)^1/p].

Indeed, the inequality (3.6) is equivalent to

(3.7) Tr[X(X^−p/2Y^pX^−p/2)^λ/p]≤Tr[X^1−λY^λ].

Then we have many counter-examples. If we set p = 0.3, λ = 0.9 and X = 10 3

3 9

, Y =

5 4 4 5

,then inequality (3.7) fails. (R.H.S. minus L.H.S. of (3.7) approximately becomes -0.00309808.) Thus, inequality (3.6) is not true in general.

Corollary 3.4.

(i) For positive matricesX andY, the trace inequality

D_λ(X|Y)≤ −Tr[Xln_λ(X^−1/2Y X^−1/2)]

holds.

(ii) For positive matricesX andY, andp≥1, we have inequality (3.1).

Proof.

(i) Putp= 1in (1) of Theorem3.3.

(ii) Take the limit asλ→0.

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References

[1] S. ABE, Heat and entropy in nonextensive thermodynamics: transmutation from Tsallis theory to Rényi-entropy-based theory,Physica A, 300 (2001), 417–

423.

[2] S. ABE, S. MART ´NEZ, F. PENNINI,ANDA. PLASTINO, Nonextensive ther- modynamic relations, Phys. Lett. A, 281 (2001), 126–130.

[3] H. ARAKI, On an inequality of Lieb and Thirring, Lett. Math. Phys., 19 (1990), 167–170.

[4] S. FURUICHI, Trace inequalities in nonextensive statistical mechanics, Linear Algebra Appl., 418 (2006), 821–827.

[5] S. FURUICHI, K. YANAGI ANDK. KURIYAMA, Fundamental properties of Tsallis relative entropy, J. Math. Phys., 45 (2004), 4868–4877.

[6] S. FURUICHI, K. YANAGIANDK. KURIYAMA, A note on operator inequal- ities of Tsallis relative opeartor entropy, Linear Algebra Appl., 407 (2005), 19–31.

[7] S. GOLDEN, Lower bounds for the Helmholtz function, Phys. Rev., 137 (1965), B1127–B1128.

[8] F.HIAI AND D.PETZ, The Golden-Thompson trace inequality is comple- mented, Linear Algebra Appl., 181 (1993), 153–185.

[9] S. MARTINEZ, F. NICOLÁS, F. PENNINIA AND A. PLASTINOA, Tsallis’

entropy maximization procedure revisited, Physica A, 286 (2000), 489–502.

[10] C. TSALLIS, Possible generalization of Bolzmann-Gibbs statistics, J. Stat.

Phys., 52 (1988), 479–487.

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[11] C. TSALLIS, R.S. MENDESCANDA.R. PLASTINO, The role of constraints within generalized nonextensive statistics, Physica A, 261 (1998), 534–554.

[12] W. THIRRING, Quantum Mechanics of Large Systems, Springer-Verlag, 1980.

[13] C.J. THOMPSON, Inequality with applications in statistical mechanics, J.

Math. Phys., 6 (1965), 1812–1813.

[14] H. UMEGAKI, Conditional expectation in an operator algebra, IV (entropy and information), Kodai Math. Sem. Rep., 14 (1962), 59–85.

[15] H. UMEGAKI AND M. OHYA, Quantum Mechanical Entropy, Kyoritsu Pub.,1984 (in Japanese).

[16] J. von NEUMANN, Thermodynamik quantenmechanischer Gesamtheiten, Göttinger Nachrichen, 273-291 (1927).

[17] K. YANAGI, K. KURIYAMA AND S. FURUICHI, Generalized Shannon in- equalities based on Tsallis relative operator entropy, Linear Algebra Appl., 394 (2005), 109–118.